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Bipartite Graphs as Models of Population Structures inEvolutionary Multiplayer GamesJorge Pena*¤a, Yannick Rochat¤b

Institute of Applied Mathematics, Faculty of Social and Political Sciences, University of Lausanne, Lausanne, Switzerland

Abstract

By combining evolutionary game theory and graph theory, ‘‘games on graphs’’ study the evolutionary dynamics offrequency-dependent selection in population structures modeled as geographical or social networks. Networks are usuallyrepresented by means of unipartite graphs, and social interactions by two-person games such as the famous prisoner’sdilemma. Unipartite graphs have also been used for modeling interactions going beyond pairwise interactions. In thispaper, we argue that bipartite graphs are a better alternative to unipartite graphs for describing population structures inevolutionary multiplayer games. To illustrate this point, we make use of bipartite graphs to investigate, by means ofcomputer simulations, the evolution of cooperation under the conventional and the distributed N-person prisoner’sdilemma. We show that several implicit assumptions arising from the standard approach based on unipartite graphs (suchas the definition of replacement neighborhoods, the intertwining of individual and group diversity, and the large overlap ofinteraction neighborhoods) can have a large impact on the resulting evolutionary dynamics. Our work provides a clearexample of the importance of construction procedures in games on graphs, of the suitability of bigraphs and hypergraphsfor computational modeling, and of the importance of concepts from social network analysis such as centrality,centralization and bipartite clustering for the understanding of dynamical processes occurring on networked populationstructures.

Citation: Pena J, Rochat Y (2012) Bipartite Graphs as Models of Population Structures in Evolutionary Multiplayer Games. PLoS ONE 7(9): e44514. doi:10.1371/journal.pone.0044514

Editor: Angel Sanchez, Universidad Carlos III de Madrid, Spain

Received July 10, 2012; Accepted August 8, 2012; Published September 10, 2012

Copyright: � 2012 Pena, Rochat. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permitsunrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Funding: These authors have no support or funding to report.

Competing Interests: The authors have declared that no competing interests exist.

* E-mail: [email protected]

¤a Current address: Faculty of Business and Economics, University of Basel, Basel, Switzerland¤b Current address: Institute of Psychology, Faculty of Social and Political Sciences, University of Lausanne, Lausanne, Switzerland

Introduction

Since the pioneering work of Maynard Smith and Price [1],

evolutionary game theory [2] has become a valuable tool to

describe and study evolutionary dynamics when fitness is

frequency-dependent. Evolutionary game theory builds on the

theory of games [3] by considering populations of individuals

whose success or fitness depends on the outcome of social

interactions. Behavioral strategies are genetically or culturally

inherited, so that the relative abundance of fitter strategies

increases over time due to natural selection or social learning.

When populations are assumed to be infinite and well-mixed, the

replicator dynamics [4,5] offers a deterministic and exact account

of the evolutionary dynamics.

In spite of the importance of the replicator dynamics as a

mathematical tool for investigating evolutionary dynamics, it is

obvious that real populations are never infinite nor perfectly well-

mixed. Games on graphs (see Refs. [6] and [7] for reviews) go

beyond these two simplifying assumptions by considering finite-

sized populations embedded in graphs representing geographical

isolation or social networks. A graph G~(V ,E) consists of a set V

of vertices and a set E of edges connecting pairs of vertices. In

general models of games on graphs, individuals are placed on two

graphs with the same set of vertices [8]: the interaction graph

G~(V ,EG) and the replacement graph H~(V ,EH ). Evolution-

ary dynamics are specified so that, first, individuals play two-

person games with their neighbors in the interaction graph G, and

second, strategy updating takes place along the edges of the

replacement graph H. Although the set of edges of the

replacement graph may differ from the set of edges of the

interaction graph, it is usually assumed that EH~EG so that G

and H effectively coincide.

A perusal of the vast literature on games on graphs highlights

the importance of network structure in the evolutionary dynamics

of different games. Two particular results are worth mentioning.

First, although unconditional cooperation under the one-shot

prisoner’s dilemma (PD) is not evolutionarily stable in infinite and

well-mixed populations, it can be viable in sparse homogeneous

networks [9–11]. ‘‘Spatial reciprocity’’ [12], ‘‘network reciprocity’’

[13] and ‘‘graph selection’’ [14] are different labels that have been

coined in order to contrast such effect with other cooperation-

promoting mechanisms (see, however, Refs. [11,15–17] for the

close connections between network reciprocity and kin selection

via limited dispersal, and between games on graphs and inclusive

fitness theory). Second, heterogeneous population structures such

as scale-free networks [18] can significantly promote cooperation

under the PD and other social dilemmas [19,20], although such

promotion strongly depends on several details of the network, the

payoff functions and the updating rules [7,21–26].

PLOS ONE | www.plosone.org 1 September 2012 | Volume 7 | Issue 9 | e44514

Notwithstanding the importance of pairwise social interactions,

many situations in real social systems require the collective action

of groups comprised by more than two individuals. Moreover,

interactions within these larger groups can not always be

represented as disjointed collections of two-person games [27].

Public goods games (PGGs) are paradigmatic among such non-

decomposable multiplayer games. PGGs are models of situations

where individuals face the dilemma of providing and/or main-

taining a public good: a common resource that is both non-

excludable (no individual can be excluded from its consumption)

and non-rivalrous (one individual’s use of the public good does not

diminish its availability to another individual) [28]. Digestive

enzymes in yeast [29], ATP in heterotrophic microorganisms [30],

webs in social spiders [31], alarm calls in meerkats [32], collective

hunting in lions [33], and open-source software in contemporary

humans [34] are typical examples of public goods whose abusive

exploitation by non-contributing individuals may lead to the so-

called tragedy of the commons [35]: a situation in which nobody

contributes and therefore no public good is produced or

maintained.

By far, the most well known PGG is the N-person prisoner’s

dilemma (NPD) [36]. In this game, each individual in a group of

size N has to decide whether to cooperate (by contributing to a

common pot) or to defect (by refraining from contributing). The

sum of the individual contributions is multiplied by a factor r and

then equally distributed among all players, including those who

did not contribute. No matter the decisions taken by the other

players, it is always better to defect if 1vrvN. In infinite well-

mixed populations and in the absence of cooperation-promoting

mechanisms, defection is evolutionarily stable and the replicator

dynamics predicts the ultimate extinction of Cs. However, as it is

also the case in the two-person PD, cooperation in the NPD can be

sustained in structured populations under particular life-cycle

assumptions. Hauert et al. [37] studied a spatial NPD resulting

from placing the individuals in the nodes of a two-dimensional

lattice and restricting interactions to nearest neighbors. In this

model and for large values of r (but still for rvN ) Cs are able to

survive by minimizing interactions with Ds through cluster

formation. Hauert et al.’s model has been extended by Santos et

al. [38], who used scale-free networks instead of regular lattices as

population structures. The highly heterogeneous degree distribu-

tions of scale-free networks introduce social diversity both at the

individual level (players vary greatly with respect to the number of

games they take part in) and at the group level (different games are

played by different numbers of players). Social diversity brings up

a moderate promotion of cooperation when Cs pay a fixed cost cper game, but a significant boost when Cs pay a fixed cost c for all

the games they play. In the following we use the terminology

introduced in Ref. [39] and call conventional NPD the former case

and distributed NPD the latter.

The way networks are constructed is a common feature of Refs.

[37,38] and many other papers dealing with evolutionary

multiplayer games on networks [40–52]. We refer to this

construction procedure as the graph approach. According to this

framework, nodes of a graph G~(V ,E) define both individuals

playing a game and games being played by the focal individual

plus its direct neighbors, so that an individual with z neighbors

takes part in zz1 games: the one centered on itself plus z games,

each centered on one of its neighbors. Fitness or social success is

given by the sum of payoffs collected in these zz1 games, and

competition or imitation takes place along the edges of the graph.

An alternative way of looking at the population structure

resulting from the graph approach is realizing that while the

replacement graph is the original graph G, the interaction graph is

actually a hypergraph (or a bipartite graph) in which hyperedges

(or top vertices) correspond to closed neighborhoods of G (see

Figure 1). A hypergraph is the generalization of a graph for the

case where edges (called in this case hyperedges) can connect

arbitrarily many vertices. A bipartite graph B~(T ,\,E), also

called a bigraph, consists of two disjoint sets of vertices, T (top

vertices) and \ (bottom vertices), and a set of edges, E. The

difference between bipartite graphs and standard unipartite graphs

is that edges in a bigraph only connect vertices of different kinds.

Undirected hypergraphs and bigraphs are mathematically equiv-

alent, but bigraphs are usually easier to implement and to work

with. Many real biological and social networks display a natural

bipartite structure and can be represented as bigraphs in a

straightforward manner. Food webs [53] and metabolic networks

[54] are well known biological examples; social examples include

affiliation [55] or collaboration networks [56], such as those

connecting co-owners of companies [57], film actors [58] and

scientists [56]. In this paper, we represent groups/games as top

vertices and individuals/players as bottom vertices. Three network

statistics will be particularly important. First, the top degree

distribution gives the distribution of the number of games being

played by a given individual. Second, the bottom degree distribution

gives the distribution of the number of individuals playing a given

game. Finally, the bipartite clustering coefficient captures correlations

between the neighborhoods of bottom vertices, i.e. the degree to

which groups overlap.

With the previous definitions, the graph approach can be

interpreted as one in which (1) the replacement graph is defined,

and (2) the interaction bigraph is constructed from the replace-

ment graph. This approach has become the de facto standard for

modeling population structures in multiplayer games on graphs.

However, it has many important limitations. First, since both

players and games are identified with the same set of vertices, the

numbers of games and players are exactly the same, i.e. DT D~D\Din the resulting interaction bigraph. Second, and for the same

reason, the top degree distribution and the bottom degree

distribution coincide. In real systems, however, these distributions

are usually very different. In collaboration networks, for example,

the number of papers per author has been shown to follow a

power-law distribution while the number of authors per paper

generally follows an exponential distribution [56]. Third, the

graph approach automatically leads to a relatively large bipartite

clustering coefficient. Although such large coefficient seems to be

an intrinsic property of many social and biological networks [59],

its presence by default in models of games on graphs can be a

drawback if the goal is to build null models of connectivity patterns

or to study the effects of bipartite clustering. Fourth, while each

individual effectively interacts with second-order neighbors in the

original graph, strategy updating is posited to occur only between

first-order neighbors. As an example of this, consider the graphs

depicted in Panels A and B of Figure 1. Note that individual Aplays with C and D the game centered on B, but A is not

connected to C nor D in the replacement graph. Finally,

replacement graphs in the graph approach do not reflect

encounter rates between two individuals but rather assume that

all neighbors in the replacement graph are equally important.

Consider again the graphs depicted in Panels A and B of Figure 1.

On the one hand, individual B plays thrice with C (the games

centered on B, C and D) but only twice with A (the games

centered on A and B). On the other hand, individual B plays (on

average) games of smaller size with A (one two-person game

centered on A and one four-person game centered on B) and

games of larger size with C (two four-person games centered at Band D, and one three-person game centered on C). In any case,

Bipartite Graphs in Evolutionary Multiplayer Games


the replacement graph posited by the graph approach fails to take

into account these heterogeneities, since the connections between

B and A and between B and C are equally (un)weighted in this

graph.

A new modeling framework for studying networked multiplayer

games, recently proposed by Gomez Gardenes and co-workers

[60,61] and further generalized here, is free of these limitations.

We call this framework the bigraph approach. It consists in (1) defining

the interaction bigraph B~(T ,\,E) so that top vertices corre-

spond to games and bottom vertices to players, and (2) deriving the

replacement graph H~(\,EH ) as the (bottom) projection of B

(see Panels C and D of Figure 1). The bottom projection of a

bigraph B is a graph H~(\,E\) so that (i,j)[E\ if and only if

i[\ and j[\ are connected at least once to the same top vertex. In

addition to the ‘‘unweighted projection’’ (UP) considered in Ref.

[60], we also consider two weighted projections: the ‘‘unnorma-

lized weighted projection’’ (UWP) and the ‘‘normalized weighted

projection’’ (NWP). With the UWP method, the weight of the link

between two players is proportional to the number of common

games played by those players; with the NWP method, the sizes of

such groups are taken into account when calculating the weights,

so that interactions in smaller groups contribute more to the total

weight of the link than interactions in larger groups (see the

sec:methods section for details).

The bigraph approach circumvents all of the limitations

associated with the graph approach we mentioned above. Since

the interaction bigraph is defined at the outset, it can have

arbitrary numbers of games and players, different degree

distributions for games and players and (if required) relatively

low bipartite clustering coefficient. In addition, since the replace-

ment graph is obtained as the bottom projection of the interaction

graph, individuals playing together at least one game will be

connected in the replacement graph. Hence, the neighborhood of

player A in the replacement graph shown in Panel D of Figure 1

comprises all the individuals A interacts with, i.e. B,C,Df g.Finally, weighted projections take into account differences in the

interaction patterns of players and reflect such differences in the

resulting replacement graph. For instance, in the graph shown in

Panel D of Figure 1 the weights of the links between players B and

A and between players B and C are respectively given by 2 and 3,

indicating the number of common games between each pair of

players (in Panel D of Figure 1, weights are derived using the UWP

method).

In this paper, we make use of the bigraph approach to explore

the influence of different topological properties of network

structures on the evolutionary dynamics of multiplayer games.

We focus on the conventional and the distributed versions of the

NPD, as these are among the most studied evolutionary

multiplayer games on graphs. Specifically, we investigate the

effects of different assumptions on the way of specifying

replacement graphs, different top and bottom degree distributions,

and different amounts of bipartite clustering. We build interaction

bigraphs either from prescribed simple graphs using the graph

approach, or from given degree distributions using the configu-

ration model procedure (see the sec:methods section). We denote

these interaction bigraphs respectively by the labels fromgraph-X

and config-Y–Z, where X stands for the simple graph from which

the bigraph is constructed, and Y and Z stand for the degree

distributions of the bottom and the top vertices, respectively.

Replacement graphs are given either by the graph approach or by

the bigraph approach.

Results

Replacement GraphsIn the graph approach, the original graph from which the

interaction bigraph is constructed automatically determines the

replacement graph. As a result, the subset of players involved in

imitation/competition with a given individual is generally smaller

than the subset of players with whom such individual interacts.

This implicit assumption is in stark contrast with most models of

two-person games on graphs, where interaction and replacement

neighborhoods perfectly overlap. In the following, we present the

results of making interaction and replacement neighborhoods

coincide in otherwise standard models of evolutionary multiplayer

games on graphs.

Figure 2 depicts the results of the evolution of cooperation in the

conventional and the distributed NPD for population structures

with the same interaction bigraph but different replacement

graphs. We plot the cooperation level (the average fraction of Cs

for 2000 additional generations after an initial transient of 105

generations) as a function of the normalized enhancement factor

g~r=n, where n is the average degree of the top nodes of the

interaction bigraphs, i.e. n is the mean number of players per game

in the population. In each case, the population structure is built (1)

by defining a graph G of order Z (i.e. G has Z nodes) and mean

degree SzT, and (2) by constructing the interaction bigraph B from

Figure 1. Modeling population structures in evolutionary multiplayer games. The graph approach consists in first defining the originalgraph G (Panel A) and then constructing the interaction hypergraph B (Panel B) by associating a hyperedge with each closed neighborhood in G. Theinteraction hypergraph can be also represented as a bigraph (Panel C), where individuals/players are bottom vertices and groups/games are topvertices. In the graph approach the replacement graph H is assumed to be equal to the original graph G, so that interactions take place along thehyperedges of the hypergraph, but strategy updating occurs along the edges of the original graph. The alternative bigraph approach consists in firstdefining the interaction bigraph B (Panel C) and then obtaining the replacement graph H (Panel D) as the bottom projection of the interactionbigraph. Weights can be attached to the links of the replacement graph according to different heuristics (here, the ‘‘unnormalized weightedprojection’’ method is used; the width of the links is proportional to the links’ weights). The interaction bigraph can be constructed from a bipartitegraph model or following the graph approach from a simple graph G (Panel A). In this last case, the replacement graphs due to the graph approach(the original graph shown in Panel A) and to the bigraph approach (the projection of the interaction bigraph shown in Panel D) differ, the latter beingdenser.doi:10.1371/journal.pone.0044514.g001



G using the graph approach. Hence, n~m~SzTz1, where m is

the average degree of the bottom nodes in the interaction

bigraphs, i.e. the mean number of games per player in the

population. The replacement graph H is given either by G itself

(graph approach) or by the projection of B (bigraph approach). In

this last case, weights are assigned to the edges of H according to

one of three methods: UP, UWP and NWP. In any case,

individuals engage in a given number of multiplayer games

(according to their connectivity in the interaction bigraph) and

accumulate payoffs. The accumulated payoff of each player is then

associated with its fitness/success, and competition/imitation is

implemented by using a finite population analogue of the

replicator dynamics [38,62]: in social learning terms, each

individual randomly chooses a neighbor in the replacement graph

and, if the neighbor’s success is greater than its own success, it

imitates the neighbor’s strategy with a probability proportional to

the success difference (see the sec:methods section for details).

Panel A of Figure 2 shows the results for the case where G is a

ring of degree z~4. We refer to the resulting bigraph B as

fromgraph-ring. Referring to the original graph G, individuals

interact with both their first-order neighbors and their second-

order neighbors. If the replacement graph is given by the graph

approach, only first-order neighbors in G are considered for

competition/imitation. If the replacement graph is given by the

bigraph approach, both first-order and second-order neighbors in

G are considered for competition/imitation, possibly with a

probability depending on the number of common games (UWP).

Note that the larger replacement neighborhoods due to the

bigraph approach favors cooperation slightly, but systematically. A

detailed analysis of the origin of such promotion, considering the

case of two contiguous clusters of Ds and Cs in a ring of degree

z~4, can be found in section 1 of Text S1 and Figure S1.

While the larger replacement neighborhoods brought about by

the bigraph approach are beneficial to cooperation in bigraphs

constructed from rings, they are detrimental to cooperation in

bigraphs constructed from Barabasi-Albert (BA) scale-free net-

works, which we call fromgraph-ba. Indeed, as evidenced in Panels

B and C of Figure 2, in this case there is systematically less

cooperation if replacement neighborhoods coincide with interac-

tion neighborhoods (bigraph approach) than if the original graph

is taken as the replacement graph (graph approach). Additionally,

in the former case the assignment of weights to the edges of the

replacement graph plays a key role in BA networks, as it is evident

from the ordering of the curves, with NWP leading to more

cooperation than UWP, and UWP to more cooperation than UP.

In order to explain these results, let us briefly recall the

mechanism responsible for the promotion of cooperation in the

distributed NPD when the interaction and replacement graphs are

derived from scale-free networks using the graph approach [38].

Scale-free networks are characterized by the co-existence of few

hubs (very well connected individuals) with a vast majority of

leaves (poorly connected individuals). Due to their large connec-

tivity, hubs not only take part in many games, consequently

accumulating high payoffs, but are also often targeted for

competition/imitation by their neighbors. As a result of these

two factors, C-hubs and D-hubs easily spread their strategies to

their less connected neighbors. However, while C-hubs are favored

by a positive feedback mechanism (the more they are imitated, the

more Cs in their neighborhoods, and the more their own

accumulated payoffs increase) D-hubs are penalized by a negative

feedback mechanism (the more they are imitated, the more Ds in

his neighborhood, and the more their own accumulated payoffs

decrease) that eventually leads to their own demise. Hubs’ inherent

success along with the feedback mechanisms favoring Cs in inter-

hub competition have been studied using star and double-star

graphs as simple models of connectivity patterns in scale-free

networks [38,39].

If the replacement graph H is no longer the original graph G

(graph approach) but it is rather assumed to be the projection of

the interaction bigraph B (bigraph approach), many additional

links are present in H that were not in G. Indeed, since each top

node of degree z induces a clique consisting of z(z{1)=2 edges,

the projection of B is a relatively dense graph, particularly if the

top degree distribution is highly heterogeneous [63]. This higher

density of the replacement graph is at the origin of the hindering of

the evolution of cooperation when moving from the graph

approach (G taken as H ) to the bigraph approach (the projection

of B taken as H ). Figures 3 and 4 show this effect for bigraphs built

according to the graph approach from star and double-star graphs.

In stars, and when the replacement graph is given by the

projection of the interaction bigraph, leaves get connected to each

other so that H is now a complete graph (see Panel D of Figure 3).

Figure 2. Cooperation level for population structures with different replacement graphs. Results are shown for the conventional NPD(Panels A and B) and the distributed NPD (Panel C). In each case, interaction graphs are constructed following the graph approach from rings (PanelA) or Barabasi-Albert scale-free networks (Panels B and C) of order Z~512 and mean degree SzT~4. The replacement graph is given by the originalgraph from which the interaction bigraph is constructed (OG) or the unweighted, unnormalized weighted or normalized weighted projection of theinteraction graph (UP, UWP and NWP, respectively).doi:10.1371/journal.pone.0044514.g002



This hinders the spreading of cooperative behavior from a C-

center when defective leaves earn a higher payoff than cooperative

leaves. In double-star graphs, leaves of the same star get

interconnected and the center of one star gets connected to the

leaves of the other star (see Panel D of Figure 4). This increased

interconnection hinders cooperation by partially destroying both

the positive feedback around C-centers and the negative feedback

around D-centers on which inter-hub competition is based in the

model of Ref. [38]. Note that, in all cases, the magnitude of these

unfavorable effects depends on the weights attached to the links of

the replacement graph. Indeed, different projection methods lead

to different weight distributions, which in turn affect the

topological importance of different nodes in the evolutionary

process. Such topological importance can be captured by what we

call in this paper replacement centrality, which we define as the

expected number of times a given node/individual is selected for

competition/imitation by its neighbors (see section 4 of Text S1).

Other things being equal, nodes with a higher replacement

centrality play a more influential role in the evolutionary

dynamics. We find that the level of centralization of the

replacement graph (defined as the degree to which a single node

is more central than others in the network; see section 4 of Text

S1) correlates with the amount of cooperation exhibited in these

topologies, as measured by the inverse fixation time of a single C-

center in a star graph (see Panel E of Figure 3) or by the

cooperation level in BA scale-free networks (Panels B and C of

Figure 2). Figure S2 shows that the relationship between projection

method and centralization of the network found in star graphs

(OG w NWP w WP w UP) is maintained in BA scale-free

networks. As evidenced by Figure S2 and Panels B and C of

Figure 2, weight distributions leading to more centralized

replacement graphs are also responsible for higher cooperation

levels.

Degree DistributionsIn bigraphs constructed using the graph approach, group diversity

(heterogeneity in the number of players per game) is inextricably

intertwined with individual diversity (heterogeneity in the number of

games per player). Indeed, the top degree distribution (determin-

ing group diversity) is exactly the same as the bottom degree

distribution (determining individual diversity) in bigraphs built

using the graph approach. In order to analyze group diversity and

individual diversity independently of each other, we made use of

random configuration model bigraphs (for which the degree

sequences of top and bottom vertices can be specified indepen-

dently of each other) as interaction bigraphs. We used two

different degree sequences for top and bottom vertices: a constant

sequence (all degrees are the same) and the degree sequence of a

BA scale-free graph, which approximately follows a power-law.

Combinations of these two degree sequences resulted in four

bigraphs: config-reg-reg (with homogeneous top and bottom

degree distributions), config-ba-reg (with heterogeneous bottom

and homogeneous top degree distributions), config-reg-ba (with

homogeneous bottom and heterogeneous top degree distributions),

and config-ba-ba (with heterogeneous bottom and top degree

distributions). The reason for using the degree sequence of a BA

graph instead of determining the degree sequence by another

method (for instance, by sampling the sequence from a random

variable distributed according to a power-law distribution) is to be

able to compare the results obtained for config-ba-ba with those

obtained for fromgraph-ba in the subsec:replacement subsection.

Indeed, config-ba-ba has the same top and bottom degree

Figure 3. Evolutionary dynamics on stars. Consider a star graph G (Panel A) consisting of one C-center connected to Z{1 leaves (m of whichare Cs) and the resulting interaction hypergraph B (Panel B) constructed from G using the graph approach. We assume that social interactions aremodeled by the distributed NPD. In the graph approach, G is taken as the replacement graph (Panel C). In this case, competition/imitation occursonly between the center and the leaves. The C-center invades D-leaves for values of r above a critical value which reduces to a~2=(1{2=Z) if m~0(the C-center is the only C). In the bigraph approach, the replacement graph is given by the projection of the interaction bigraph, so that leaves arenow interconnected and the resulting topology is no longer a star but a complete graph (Panel D). The creation of these new links allows for inter-leaf competition/imitation, which is favorable to Ds if rv4. As a result, for avrv4, the time to fixation to the absorbing state where all individuals areCs can become arbitrarily large depending on the weights attached to the links of the replacement graph, as it is shown in Panel E for Z~10, r~2:8and for replacement graphs given by the OG, NWP, UWP and the UP methods. Panel F shows these replacement graphs together with the values ofthe weights of the links (whl for the weight of the link between the center and a leaf; wll for the weight of the link between two leaves) and theircentralization indices (rX ). Note that more centralized graphs correspond to those more favorable to the spreading of cooperative behavior from thecenter. See section 2 of Text S1 for the derivation of the invasion conditions shown in Panels C and D, and sections 4 and 5 of Text S1 for thederivation of the centralization indices shown in Panel F.doi:10.1371/journal.pone.0044514.g003



sequences as fromgraph-ba, and can be effectively thought of as a

randomization of such network.

Figure 5 shows the results for the evolution of cooperation in the

conventional and the distributed NPD for the four configuration

model bigraphs and for the fromgraph-ba. Let us consider first the

results for config-reg-reg, i.e. the homogeneous population

structure lacking social diversity of any kind. As shown in the

figure, this network is able to sustain cooperation for values of gabove gc&0:7. Furthermore, cooperation is fully established for

gwgd , with gd close to its value for infinite well-mixed populations

(gd~1). For gcvgvgd , Cs and Ds co-exist in dynamical

equilibrium. If group diversity is introduced (config-reg-ba), the

co-existence zone grows so that gc&0:6, and gdw1:2. This shows

that group diversity has mixed effects in the evolutionary

dynamics, promoting cooperation (with respect to config-reg-reg)

up to a critical value g�&0:85, and hindering cooperation above

this value. If diversity is instead introduced at the individual level

(config-ba-reg), cooperation is evolutionarily viable for gw0:55 in

the conventional NPD and for gw0:45 in the distributed NPD.

Note, nonetheless, that defective behavior is not completely

eradicated, not even for gw1. From these results, it is evident

that individual diversity leads to higher cooperation levels than

group diversity (compare the curves for config-ba-reg with those

for config-reg-ba) for all values of g. We also note that the levels of

cooperation slightly improve when both kinds of social diversity

are simultaneously present (compare config-ba-ba to config-ba-reg

and config-reg-ba). Finally, the results obtained with config-ba-ba

are almost the same as those obtained with fromgraph-ba, which

suggests that the higher topological correlations present in

fromgraph-ba and absent in config-ba-ba play a rather small role

in the evolutionary dynamics.

The results for networks with homogeneous bottom degree

distributions (config-reg-reg and config-reg-ba) and for networks

with heterogeneous bottom degree distributions (config-ba-reg,

config-ba-ba and fromgraph-ba-ba) differ not only quantitatively

in their cooperation levels, but also qualitatively in their dynamics.

Indeed, intermediate cooperation levels for bigraphs with homo-

geneous bottom degree distributions are mostly due to the co-

existence of Cs and Ds. Contrastingly, in the case of bigraphs with

heterogeneous bottom degree distributions intermediate coopera-

tion levels are due to bi-stability, so that the vast majority of times

the dynamics reaches the absorbing states of full defection or full

cooperation. In this last case, intermediate cooperation levels are

Figure 4. Evolutionary dynamics on double stars. Consider a double-star graph G (Panel A) consisting of a left and a right star connected bythe centers, and the resulting interaction hypergraph B (Panel B) constructed following the graph approach. In the graph approach, G is taken as thereplacement graph (Panel C). In this case, and for a wide range of values of r, spreading occurs preferentially from the centers (or hubs) to theirrespective leaves. Long-term evolution will ultimately depend on inter-hub competition, which is favorable to C-hubs due to the positive andnegative feedback mechanisms brought about by the spreading from centers to (own) leaves. In the bigraph approach, the replacement graph isgiven by the projection of the interaction bigraph (Panel D), so that the center of one start gets connected to the leaves of the other star and leavesof the same star get connected with each other. This not only allows successful centers to breed copies of themselves in the leaves of the other star,but also makes inter-leaf competition possible, which is favorable to Ds if rv4. As a result, the feedback mechanisms on which the evolution ofcooperation on heterogeneous graphs is based are diminished and the evolutionary outcome is more favorable to Ds. This is illustrated in Panels Eand F, which show typical scenarios for the time evolution of the fraction of Cs under the distributed NPD (r~1:3) on the leaves of double-star graphs(Panel E: X~10, Y~20; Panel F: X~20, Y~10), for replacement graphs given by the OG, UP, UWP and NWP methods. In all cases we placed Cs onall nodes of the double-star, except for the left center, where we placed a D (see configurations a and e of Panel G). If the replacement graph is givenby the original graph (OG), the dynamics are such that, typically, the D-center invades the leaves of his star (configuration b), then the C-centerinvades the D-center (c) and finally D-leaves on the left star are invaded by the C-center (d). When the replacement graph is given by the projectionof the interaction graph, Ds can now easily spread from the initial center until they invade the whole population (e, f, g). Weights attached to thelinks of the projection play a key role in this case, with NWP still favoring Cs when the connectivity of the left star is small compared to that of thesecond star (e, f, h, i). See section 3 of Text S1 for the analytical derivation of the results shown in this figure.doi:10.1371/journal.pone.0044514.g004



almost entirely determined by the proportion of times the

dynamics ended up in the full cooperation absorbing state.

Figure 6 provides some insight on the different results obtained

when diversity is introduced at the individual level (config-ba-reg)

or at the group level (config-reg-ba). First, note that the degree

distribution of the replacement graph for config-ba-reg is highly

heterogeneous (see top panels of Figure 6). Indeed, it is well known

that the degree distribution of the projection of a bigraph with a

power law bottom degree distribution also follows a power law

[64–66]. Contrastingly, the degree distribution of the replacement

graph for config-ba-reg is less heterogeneous. A second important

difference between config-ba-reg and config-reg-ba is the way

received benefits are distributed on these networks. When the

population consists of 50% Cs randomly placed on the bottom

vertices of the bigraph, the distribution of received benefits closely

follows a power-law in the case of config-ba-reg, but it

approximately follows a normal distribution in the case of

config-reg-ba (see middle panels of Figure 6). The reason behind

these different distributions is that on config-ba-reg both the per-

capita per-game contribution and the number of games per

individual are highly variable while on config-reg-ba they are

constant. This leads to a highly heterogeneous distribution of

received benefits on config-reg-ba and to a relatively homogeneous

distribution on config-reg-ba. Finally, while there is a strong

correlation between connectivity in the replacement graph and

received benefit in config-ba-reg, such correlation is practically

inexistent in config-reg-ba (see bottom panels of Figure 6). Indeed,

for config-ba-reg hubs in the replacement graph are individuals

participating in many games and hence accumulating large

payoffs. Contrastingly, for config-reg-ba highly connected indi-

viduals in the replacement graphs are those participating in large

groups, which have on average the same proportion of Cs and hence

produce the same amount of public good than smaller groups. As a

result, the evolutionary dynamics on config-ba-reg is dominated by

a small number of very well connected and powerful individuals,

while config-reg-ba is far more homogeneous, both concerning

connectivity in the replacement graph and accumulated payoffs.

These differences translate into two different modes of evolution.

In config-ba-reg (see Figure S3) the influence of hubs is decisive to

the evolutionary outcome, so that a majority of C-hubs leads the

whole population to the all-Cs absorbing state, while a majority of

D-hubs leads the population to the all-Ds absorbing state.

Additionally, the proportion of Cs is also higher in high-degree

classes (very well connected individuals) than in low-degree classes

(poorly connected individuals). Contrastingly, in config-reg-ba (see

Figure S4) the evolutionary dynamics is largely independent of

what happens with well-connected individuals, and evolution

unfolds as a process of dynamical self-organization in which Cs

tend to cluster in small groups which are more favorable to

cooperation while Ds tend to do so in large groups which are more

favorable to defection.

Bipartite Clustering CoefficientThe bipartite clustering coefficient captures the degree to which

bottom vertices’ neighborhoods overlap (see section 6 of Text S1

for details). As pointed out in the sec:introduction section,

interaction bigraphs built using the graph approach lead, by

construction, to relatively high bipartite clustering coefficients. In

order to assess the real importance of clustering in the evolutionary

dynamics, we considered four interaction bigraphs with the same

top and bottom degree distributions (regular sequences in all cases)

but different bipartite clustering coefficients: fromgraph-ring

(constructed from a ring network of degree z~4), fromgraph-reg

(constructed from a random regular network of degree z~4),

fromgraph-vn (constructed from a square lattice with a von

Neumann neighborhood), and config-reg-reg (random configura-

tion model with regular top and bottom degree sequences).

Figure 7 shows the cooperation levels under the conventional

NPD and Figure 8 the bipartite clustering coefficient and the mean

degree of the replacement graph for these different bigraphs.

Interestingly, bigraphs with more bipartite clustering (and

consequently lower mean degree in the replacement graph) lead

in general to equal or higher cooperation levels for all the

considered values of the normalized enhancement parameter g.

These results make sense in the light of well established results on

Figure 5. Cooperation level for population structures with different degree distributions. Panel A shows results for the conventionalNPD; Panel B for the distributed NPD. config-X–Y stands for a configuration model bigraph with a degree sequence of type X for the bottom vertices(players) and of type Y for the top vertices (games). For the degree sequences themselves, reg is a regular sequence and ba is the degree sequence ofa Barabasi-Albert network. A bigraph constructed from a Barabasi-Albert network following the graph approach (fromgraph-ba) is shown forcomparison purposes. Parameters: m~n~5, Z~512, replacement graph given by the NWP method.doi:10.1371/journal.pone.0044514.g005



the effects of local interactions on the evolutionary dynamics of the

pairwise and multiplayer versions of the NPD. It is well known that

spatial structure enables Cs to form clusters within which they

preferentially interact with other Cs, thus reducing the exploitation

by surrounding Ds. Cluster formation is brought about by a

feedback mechanism resulting from imitation/competition with

direct neighbors that amplifies initial inhomogeneities in the

distribution of strategies. As it is shown in Figure S5, large values

of bipartite clustering coefficient favor cluster formation by

allowing Cs to find each other more easily and to reduce the

number of connections with surrounding Ds.

Discussion

Since the seminal works by Axelrod [67] and Nowak and May

[9] on the evolution of cooperation on lattices, games on graphs

have traditionally made use of unipartite graphs in order to model

population structures. Despite its usefulness for exploring the

effects of local interactions on the evolutionary dynamics of two-

player games, the use of unipartite graphs as population structures

entails a certain number of construction limitations when applied

to general multiplayer games, leading not only to a lack of

flexibility but also to unrealistic assumptions about the topological

properties of networked populations. In this paper, we have shown

how the use of bipartite graphs and of constructing procedures

that fully take into account the bipartite nature of social and

biological populations can circumvent the limitations of the

standard graph approach, opening up new opportunities for

studying the role of different properties of network topologies on

the evolution of strategic interactions. In particular, it is important

to emphasize the need of explicitly defining two graphs: the

interaction bigraph, determining who plays with whom, and the

replacement graph, determining who competes with whom. As

demonstrated in this paper, different ways of constructing any of

these two graphs or of deriving one from the other can have

important consequences in the evolutionary dynamics of multi-

player games.

First, the implicit assumption that the replacement graph

coincides with the original graph in the graph approach is crucial

for the success of BA scale-free networks as cooperation-promoting

topologies reported in Ref. [38]. When the replacement graph is

derived in a more natural way, so that interaction and replacement

neighborhoods perfectly overlap (the usual assumption in evolu-

tionary two-person games on networks) cooperation is hindered in

BA scale-free networks to a point that any advantage of social

heterogeneity is effectively canceled by the resulting larger

replacement neighborhoods (see Figure 2). The introduction of

weights in the replacement graph somewhat alleviates this

problem, as weighted links partly restore the high centralization

characteristic of BA scale-free networks.

Second, while individual diversity (heterogeneous bottom

degree distributions) systematically fosters cooperation, group

diversity (heterogeneous top degree distributions) promotes coop-

eration up to a critical value of the enhancement factor, but

hinders cooperation above such value (see Figure 5). We also

showed that networks with both kinds of social diversity foster

more cooperation than networks with only one kind of diversity,

but that the difference between the cooperation levels of networks

with both individual and group diversity and the cooperation

levels of networks with only individual diversity are relatively

small. Finally, intermediate cooperation levels in networks without

individual diversity are mostly due to co-existence of Ds and Cs,

while intermediate cooperation levels in networks with individual

diversity are characterized by bi-stable evolutionary dynamics. In

other words, the results for config-reg-reg and config-reg-ba shown

in Figure 5 can be better understood as representing the final

proportion of Cs in a population where both Cs and Ds are

present. Contrastingly, the results for config-ba-ba, config-ba-reg

and fromgraph-ba can be better interpreted as a probability of

ending up in a fully cooperative state when starting from a

condition where 50% Cs are randomly placed on the network.

Third, bipartite clustering, i.e. group overlap, plays an

important role in the evolution of cooperation under the

conventional NPD. We provided clear evidence of the beneficial

role of bipartite clustering on cluster formation and, consequently,

on the evolution of cooperation on regular structures. In this

respect, our results mirror similar conclusions on the beneficial

effects of unipartite clustering on the evolution of cooperation

under the standard evolutionary two-player PD [68–70].

Figure 6. Statistics for config-ba-reg and config-reg-ba. The figure shows some statistics for config-ba-reg (left panels) and config-reg-ba(right panels). Top panels: degree distribution of the replacement graph. Middle panels: histograms for the received benefit. The received benefit iscalculated as the payoff for Ds when approximately half of the population are Cs (randomly distributed) under the distributed NPD. Bottom panels:smooth scatter plots, regression lines and Pearson’s correlation coefficients for the received benefit vs. degree in the replacement graph. Parameters:Z~512, m~n~5 and g~0:7. The figures show statistics for 100 randomly generated networks of each type.doi:10.1371/journal.pone.0044514.g006

Figure 7. Cooperation level for bigraphs with differentbipartite clustering coefficients. The interaction bigraphs areconstructed following the graph approach with a ring (fromgraph-ring), a square lattice with von Neumann neighborhoods (fromgraph-vn), or a regular random network (fromgraph-reg) of degree z~4 asoriginal graphs, or given by a configuration model bigraph with regulardegree sequences for both top and bottom vertices (config-reg-reg). Inall four cases the degree distributions of top and bottom vertices is aregular sequence with m~n~5, the replacement graph is given by thenormalized weighted projection (NWP) of the interaction bigraph, andZ~1024.doi:10.1371/journal.pone.0044514.g007



Apart from the present paper and to the best of our knowledge,

only two studies have made use of the bigraph approach for

studying evolutionary multiplayer games: Ref. [60], where the use

of bigraphs as population structures for evolutionary games on

networks was first introduced, and Ref. [61], a subsequent study

on the effects of social diversity on the evolution of cooperation

under the NPD. In the first of these studies, the evolution of

cooperation under the NPD on a real bipartite collaboration

network is compared to the dynamics on its bottom projection.

Higher cooperation levels are found for the bipartite network than

for its projection. These results have been interpreted as hinting

that ‘‘the intrinsic group structure (described by means of the

bipartite graph) promotes cooperation in PGGs, this being a new

mechanism for this phenomenon beyond the scale-free character

and other features of the one-mode (projected) complex network’’

[60]. We would like to point out that a simpler explanation is that,

by construction, the mean group size in the bigraph built from a

projected network is always larger than the mean group size in the

original bipartite network, and that larger group sizes hinder the

evolution of cooperation under the NPD. In order to assess the

influence of group structure and other mesoscopic properties on

the evolutionary dynamics, a comparison of real bipartite networks

with their ‘‘randomized’’ versions should be carried out, as it has

been done for real unipartite networks and two-person games [71].

In the second study (Ref. [61]) the evolutionary dynamics of the

conventional and distributed versions of the NPD were investi-

gated on interaction bigraphs with tunable individual diversity but

no group diversity at all. The main finding of this study is that

bigraphs with low individual diversity (Poisson-like bottom degree

distributions) can actually allow for more cooperation than

bigraphs with high individual diversity (bottom degree distribu-

tions following a power law) in the case of the conventional NPD.

This result contrasts sharply with our own results, which suggest

that individual diversity generally promotes cooperation. Note,

however, that we used both a different network model (configu-

ration random networks) and different degree distributions (with

zero instead of moderate individual diversity). These different

setups could account for the divergent results. We also note that

Gomez-Gardenes et al. [61] suggest that the ability of BA scale-

free networks to outperform homogeneous networks reported in

Ref. [38] is ‘‘intrinsically due to the entanglement of social and

group heterogeneities’’. Although our own results partially support

this view, given the (moderate) synergy between individual and

group diversity, we have provided evidence that the promotion of

cooperation reported in Ref. [38] is mainly due to the implicit

assumption that the replacement graph is equal to the original

graph from which the interaction topology is constructed.

The choice of the NPD as case of study in this paper was based

on the fact that most of the theoretical work on evolutionary

multiplayer games has focused on this particular game. However,

recent empirical [29] and theoretical [72] work testifies a growing

discomfort with the NPD as model of realistic social dilemmas, in

particular because of its linearity and because of the fact that

cooperation is a strictly dominated strategy in this game. Several of

the conclusions drawn in the present study will necessarily change

if strategic interactions are modeled after PGGs different from the

NPD. For instance, it has been recently shown that, even in the

absence of a fixed topology, group diversity can importantly affect

the evolutionary dynamics of non-linear PGGs [73]. In the light of

these results, we would expect group diversity to play a more

prominent role in the evolutionary dynamics of non-linear games

played on bigraphs with highly heterogeneous top degree

Figure 8. Graphical representation and bipartite clustering coefficients of different interaction bigraphs. The figure shows typicalinteraction neighborhoods for a focal individual (red node) as well as the degree of the replacement graph (z’) and the bipartite clustering coefficient(cc.(\)) for fromgraph-ring (Panel A), fromgraph-vn (Panel B), fromgraph-reg (Panel C) and config-reg-reg (Panel D). For all networks, m~n~5. Valuesof z’ and cc.(\) are exact for fromgraph-ring and fromgraph-vn and analytical approximations (assuming networks are Bethe lattices) for fromgraph-reg and config-reg-reg.doi:10.1371/journal.pone.0044514.g008



distributions. Also, bipartite clustering could be partially detri-

mental, instead of largely beneficial, for the evolution of

cooperation if the social dilemma is modeled after a multiplayer

game with a structure similar to the snowdrift game, as it is already

the case for two-person games [62].

Methods

Population StructuresPopulation structures are modeled by means of two graphs: the

interaction bigraph B~(T,\,EB) and the replacement graph

H~(\,EH ). The two sets of vertices of the interaction bigraph (Tand \) represent, respectively, the set of groups/games and the set

of individuals/players.

Graph approach. In what we call the graph approach

[37,38], first the replacement graph H~(\,EH ) is defined, then

the interaction bigraph B~(T,\,EB) is constructed from the

replacement graph as follows. Denote by v1,v2, . . . ,vZ the vertices

of the graph H and by NH vi½ � the closed neighborhood of vertex

vi, defined as the set of vertices adjacent to vi plus vi itself. Further,

denote by b1,b2, . . . ,bZ the bottom (\) vertices of B and by

t1,t2, . . . ,tZ the top (T) vertices of B. Then, EB is defined as the set

of all pairs (bi,tj)[\|T such that vi[NH vj

� �.

Bigraph approach. In what we call the bigraph approach

[60], first the interaction bigraph B~(T,\,EB) is defined, then

the replacement graph H~(V ,EH ) is constructed by projecting

the interaction bigraph into its set of bottom vertices. In addition,

weights can be attached or not to the edges of H according to one

of the following three methods:

1. Unweighted projection (UP). As done in [60], no weights

are attached to the edges or, equivalently, the weights of all

edges have a value of one.

2. Unnormalized weighted projection (UWP). The weight

wij of the link (i,j)[EH is given by the number of games i and j

are connected to in the interaction bigraph [55]. From a social

learning perspective, the reason behind this heuristic is that the

more often i interacts with j, the better i is supposed to be

acquainted with j and therefore the more often i should

consider j as target for imitation.

3. Normalized weighted projection (NWP). The weight wij

is given by [56]

wij~Pk

dki

dkj

nk{1,

where dki ~1 if i participates in game k, dk

i ~0 otherwise, and nk is

the number of players of game k. From a social learning

perspective, the reason behind this heuristic is the assumption

that individuals get acquainted with others more easily in smaller

than in larger groups.

Bigraphs built from simple graphs using the graph

approach. For fromgraph-X interaction bigraphs, we consid-

ered four different kinds of graphs: rings, scale-free networks,

square lattices with von Neumann neighborhoods and regular

random networks. Rings are one-dimensional lattices with

degree z. Regular random networks (maximally random graphs

where each node has the same degree z) were constructed using

the igraph [74] implementation of the algorithm by Viger and

Latapy [75]. Scale-free networks were obtained by means of the

Barabasi-Albert (BA) model [18], i.e. growing networks using a

preferential attachment rule. In order to get graphs with

average degrees exactly equal to SzT, we started the growing

procedure from a fully connected graph of m0~SzTz1 nodes,

and added m~SzT=2 new edges per new node.

Configuration model bigraphs. Config-X–Y bigraphs were

constructed using the configuration model [63,64,76] with a top

degree distribution of type X and a bottom degree distribution of

type Y. For the degree distributions, we used regular sequences

(reg) and degree sequences from BA scale-free networks (ba),

constructed following the procedure mentioned before.

Multiplayer GamesEach individual i participates in all games k such that (i,k)[EB.

The social success of an individual is given by the sum of the

payoffs obtained in all games it takes part in. We considered two

versions of the NPD: the conventional NPD and the distributed

NPD [38,39]. In the conventional NPD, the payoffs of a D and a

C in a group k of size Nk are respectively given by PD~rmkc=Nk

and PC~PD{c, where mk is the number of Cs in group k, c is

the cost of cooperation and r is the enhancement factor. In the

distributed NPD, each C of degree zi (i.e. taking part in zi games)

contributes c=zi to each game, so that the overall contribution of

any C is equal to c. In this case, the payoff of individual i with

strategy si (1 if C, 0 if D) is given by [38]

Pi~X

k[NB(i)

r

zk

Xj[NB(k)

c

zj

sj

0@

1A{csi,

where NB(i) is the open neighborhood of player i in B (i.e. the set

of games played by i), NB(j) is the open neighborhood of game k

in B (i.e. the set of players participating in game j), and sj and zj

stand respectively for the strategy and the degree of the j-th player

in the k-th group.

Evolutionary DynamicsThe success/fitness of each individual was calculated as the sum

of the payoffs obtained in all the games it participates in. Strategies

are updated synchronously using a finite population analogue of

the replicator dynamics commonly used in the literature of games

on networks [38,62]. When updating the strategy of individual i, a

neighbor j of i in the replacement graph is randomly chosen with a

probability pij given by

pij~wijP

k[NH (i)

wik

,

where wij is the weight of the link (i,j)[EH . Denote by Pi the

accumulated payoff of individual i. Then, if Pi§Pj , i stays with

its current strategy; otherwise it changes its strategy to j’s with a

probability given by (Pj{Pi)=M, where M is a normalization

factor given by the highest possible difference between the

accumulated payoffs of i and j.

SimulationsSimulations were started with 50% of Cs randomly placed on

the graph. We measured the average fraction of Cs for 2000

additional generations after an initial transient of 105 generations,

and called this value the cooperation level. Data points in Figures 2

and 5 correspond to the mean cooperation level over 1000

simulations; data points in Figure 7 correspond to the mean



cooperation level over 100 simulations. A new realization of the

graph is done for each simulation.

Supporting Information

Figure S1 Evolutionary dynamics on rings. In the inset,

we plot a ring of degree z~4: the neighborhood of each node

comprises the closest two nodes to the left and to the right.

Following the graph approach, each node is the center of a game

of size five so that each individual ends up interacting with the

closest four neighbors to the left and the closest four neighbors to

the right. We assume that the initial distribution of strategies is

such that nodes iw0 are Cs and nodes iƒ0 are Ds. In the main

panel, we plot the probabilities of switching strategies for the

individuals at the boundary (nodes 0 and 1) when the replacement

graph is given by the original graph (OG) and when it is given by

the unweighted projection (UP) of the interaction bigraph. As

shown, P(s0?C)wP(s1?D)urw5=7 for the graph approach,

while P(s0?C)wP(s1?D)urw1=2 for the bigraph approach.

See section 1 of Text S1 for the calculation of these probabilities.

(TIFF)

Figure S2 Centralization of the replacement graphs forinteraction bigraphs built from Barabasi-Albert scale-free networks. Each boxplot shows the distribution of the

centralization for a random sample of 104 replacement graphs

given by the original graph (OG), the normalized weighted

projection (NWP), the unnormalized weighted projection (UWP)

and the unweighted projection (UP). In all cases, the original

graph is a Barabasi-Albert scale-free network of order Z~512 and

mean degree SzT~4. The projections are taken from bipartite

graphs constructed from the original graph using the graph

approach. Notice that more centralized networks lead to higher

cooperation levels in Panels B and C of Figure 2 in the main text.

See section 4 of Text S1 for the definition of the centralization

indices used in this figure.

(TIFF)

Figure S3 Time-dependence of the fraction of coopera-tors for different connectivity classes in the config-ba-reg network. The figure shows the fraction of Cs among low-

degree (zivm), medium-degree (mƒzivzmax=3) and high-degree

(zmax=3ƒziƒzmax) individuals, for two different simulation runs.

In Panel A, initially more than the 60% of the highly-connected

individuals are Cs. C-hubs lead the evolutionary process and

diffuse cooperative behavior among their less connected neigh-

bors. In Panel B, initially less than 40% of the hubs are Cs. Less

connected individuals quickly turn to defection, with medium-

degree and high-degree individuals eventually following the trend.

Parameters: g~0:7, m~n~5 and Z~512.

(TIFF)

Figure S4 Time-dependence of the average experiencedgroup size and of the fraction of cooperators in groups ofdifferent size for config-reg-ba. The figure shows the mean

experienced group size for Cs and Ds (top panels) and the fraction

of Cs in small (Nivn), medium-sized (nƒNivNmax=3) and large

(Nmax=3ƒNiƒNmax) groups (bottom panels) for g~0:7 (left

panels) and g~1:2 (right panels). The evolutionary dynamics on

this population structure is such that Cs preferentially cluster

together in small groups and Ds cluster together in large groups.

Parameters: m~n~5 and Z~512.

(TIFF)

Figure S5 Time evolution of the degree of assortment inthe replacement graphs of interaction bigraphs withdifferent bipartite clustering coefficients. The figure shows

the time evolution of the degree of assortment in the replacement

graph. See section 7 of Text S1 for the definition of degree of

assortment we used in this figure.

(TIFF)

Text S1 Supporting Text on analysis of the evolutionarydynamics on rings, stars and double-star graphs, and ondefinitions of replacement centrality, centralization,bipartite clustering coefficient and degree of assort-ment.

(PDF)

Acknowledgments

Discussions with Jesus Gomez-Gardenes, Marco Tomassini and Henri

Volken are gratefully acknowledged. Thanks to Isis Giraldo, editor Angel

Sanchez and one anonymous reviewer for their comments on the

manuscript and to Hamid Hussain-Khan and Andreas Huber for their

help with GridUNIL, the grid computing platform at the University of

Lausanne we used for running the simulations.

Author Contributions

Conceived and designed the experiments: JP. Performed the experiments:

JP YR. Analyzed the data: JP YR. Contributed reagents/materials/

analysis tools: JP YR. Wrote the paper: JP YR.

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